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FORMALIZED MATHEMATICS Number 1, January 1990 Université Catholique de Louvain Topological Spaces and Continuous Functions Beata Padlewska1 Warsaw University Bialystok Agata Darmochwal2 Warsaw University Bialystok Summary. The paper contains a definition of topological space. The following notions are defined: point of topological space, subset of topological space, subspace of topological space, and continuous function. The articles [5], [7], [6], [1], [4], [2], and [3] provide the terminology and notation for this paper. We consider structures TopStruct, which are systems hhcarrier , topologyii where carrier has the type DOMAIN, and topology has the type Subset-Family of the carrier. In the sequel T has the type TopStruct. The mode TopSpace , which widens to the type TopStruct, is defined by the carrier of it ∈ the topology of it & (for a being Subset-Family of the carrier of it [ st a ⊆ the topology of it holds a ∈ the topology of it) & for a,b being Subset of the carrier of it st a ∈ the topology of it & b ∈ the topology of it holds a ∩ b ∈ the topology of it . We now state a proposition (1) the carrier of T ∈ the topology of T & (for a being Subset-Family of the carrier of T 1 2 Supported by RPBP.III-24.C1. Supported by RPBP.III-24.C1. f c 223 1990 Fondation Philippe le Hodey ISSN 0777-4028 224 Beata Padlewska and Agata Darmochwal st a ⊆ the topology of T holds [ a ∈ the topology of T ) & (for p,q being Subset of the carrier of T st p ∈ the topology of T & q ∈ the topology of T holds p ∩ q ∈ the topology of T ) implies T is TopSpace . In the sequel T , S, GX will have the type TopSpace. Let us consider T . Point of T stands for Element of the carrier of T. The following proposition is true (2) for x being Element of the carrier of T holds x is Point of T. Let us consider T . Subset of T stands for set of Point of T. We now state a proposition (3) for P being Subset of the carrier of T holds P is Subset of T. In the sequel P , Q will have the type Subset of T ; p will have the type Point of T . Let us consider T . Subset-Family of T stands for Subset-Family of the carrier of T. Next we state a proposition (4) for F being Subset-Family of the carrier of T holds F is Subset-Family of T. In the sequel F will denote an object of the type Subset-Family of T . The scheme SubFamEx1 concerns a constant A that has the type TopSpace and a unary predicate P and states that the following holds ex F being Subset-Family of A st for B being Subset of A holds B ∈ F iff P[B] for all values of the parameters. One can prove the following propositions: (5) ∅ ∈ the topology of T, (6) the carrier of T ∈ the topology of T, (7) for a being Subset-Family of T [ st a ⊆ the topology of T holds a ∈ the topology of T, Topological Spaces and Continuous . . . (8) 225 P ∈ the topology of T & Q ∈ the topology of T implies P ∩ Q ∈ the topology of T. We now define two new functors. Let us consider T . The functor ∅T, with values of the type Subset of T , is defined by it = ∅ the carrier of T. The functor Ω T, with values of the type Subset of T , is defined by it = Ω the carrier of T. One can prove the following four propositions: (9) ∅T = ∅ the carrier of T, (10) Ω T = Ω the carrier of T, (11) ∅(T ) = ∅, (12) Ω (T ) = the carrier of T. Let us consider T , P . The functor P c, yields the type Subset of T and is defined by it = P c . Let us consider T , P , Q. Let us note that it makes sense to consider the following functors on restricted areas. Then P ∪Q is Subset of T, P ∩Q is Subset of T, P \Q is Subset of T, . Q P− is Subset of T. The following propositions are true: (13) p ∈ Ω (T ), (14) P ⊆ Ω (T ), 226 Beata Padlewska and Agata Darmochwal (15) P ∩ Ω (T ) = P, (16) for A being set holds A ⊆ Ω (T ) implies A is Subset of T, (17) P c = Ω (T ) \ P, (18) P ∪ P c = Ω (T ), (19) P ⊆ Q iff Q c ⊆ P c , (20) P =P cc , (21) P ⊆ Q c iff P ∩ Q = ∅, (22) Ω (T ) \ (Ω (T ) \ P ) = P, (23) P 6= Ω (T ) iff Ω (T ) \ P 6= ∅, (24) Ω (T ) \ P = Q implies Ω (T ) = P ∪ Q, (25) Ω (T ) = P ∪ Q & P ∩ Q = ∅ implies Q = Ω (T ) \ P, (26) P ∩ P c = ∅(T ), (27) Ω (T ) = (∅T ) c , (28) P \Q = P ∩Qc, (29) P = Q implies Ω (T ) \ P = Ω (T ) \ Q. Let us consider T , P . The predicate P is open is defined by P ∈ the topology of T. One can prove the following proposition (30) P is open iff P ∈ the topology of T. Let us consider T , P . The predicate P is closed is defined by Ω (T ) \ P is open . One can prove the following proposition (31) P is closed iff Ω (T ) \ P is open . Let us consider T , P . The predicate P is open closed is defined by P is open & P is closed . Topological Spaces and Continuous . . . 227 We now state a proposition P is open closed iff P is open & P is closed . (32) Let us consider T , F . Let us note that it makes sense to consider the following functor on a restricted area. Then [ F is Subset of T. Let us consider T , F . Let us note that it makes sense to consider the following functor on a restricted area. Then \ F is Subset of T. Let us consider T , F . The predicate F is a cover of T is defined by The following proposition is true F is a cover of T iff Ω (T ) = (33) Ω (T ) = [ [ F. F. Let us consider T . The mode SubSpace of T, which widens to the type TopSpace, is defined by Ω (it) ⊆ Ω (T ) & for P being Subset of it holds P ∈ the topology of it iff ex Q being Subset of T st Q ∈ the topology of T & P = Q ∩ Ω (it). Next we state two propositions: (34) Ω (S) ⊆ Ω (T ) & (for P being Subset of S holds P ∈ the topology of S iff ex Q being Subset of T st Q ∈ the topology of T & P = Q ∩ Ω (S)) implies S is SubSpace of T, (35) for V being SubSpace of T holds Ω (V ) ⊆ Ω (T ) & for P being Subset of V holds P ∈ the topology of V iff ex Q being Subset of T st Q ∈ the topology of T & P = Q ∩ Ω (V ). Let us consider T , P . Assume that the following holds P 6= ∅(T ). The functor T | P, with values of the type SubSpace of T , is defined by Ω (it) = P. 228 Beata Padlewska and Agata Darmochwal One can prove the following proposition (36) P 6= ∅(T ) implies for S being SubSpace of T holds S = T | P iff Ω (S) = P. Let us consider T , S. map of T, S stands for Function of (the carrier of T ),(the carrier of S). Next we state a proposition (37) for f being Function of the carrier of T,the carrier of S holds f is map of T, S. In the sequel f has the type map of T , S; P 1 has the type Subset of S. Let us consider T , S, f , P . Let us note that it makes sense to consider the following functor on a restricted area. Then f◦P is Subset of S. Let us consider T , S, f , P 1. Let us note that it makes sense to consider the following functor on a restricted area. Then f -1 P 1 is Subset of T. Let us consider T , S, f . The predicate f is continuous is defined by for P 1 holds P 1 is closed implies f -1 P 1 is closed . The following proposition is true (38) f is continuous iff for P 1 holds P 1 is closed implies f -1 P 1 is closed . The scheme TopAbstr concerns a constant A that has the type TopSpace and a unary predicate P and states that the following holds ex P being Subset of A st for x being Point of A holds x ∈ P iff P[x] for all values of the parameters. One can prove the following propositions: (39) for X ′ being SubSpace of GX for A being Subset of X ′ holds A is Subset of GX, (40) for A being Subset of GX, x being Any st x ∈ A holds x is Point of GX, (41) for A being Subset of GX st A 6= ∅(GX) ex x being Point of GX st x ∈ A, (42) Ω (GX) is closed , Topological Spaces and Continuous . . . (43) 229 for X ′ being SubSpace of GX, B being Subset of X ′ holds B is closed iff ex C being Subset of GX st C is closed & C ∩ (Ω (X ′ )) = B, (44) for F being Subset-Family of GX st F 6= ∅ & for A being Subset of GX st A ∈ F holds A is closed \ holds F is closed . The arguments of the notions defined below are the following: GX which is an object of the type TopSpace; A which is an object of the type Subset of GX. The functor Cl A, yields the type Subset of GX and is defined by for p being Point of GX holds p ∈ it iff for G being Subset of GX st G is open holds p ∈ G implies A ∩ G 6= ∅(GX). We now state a number of propositions: (45) for A being Subset of GX, p being Point of GX holds p ∈ Cl A iff for C being Subset of GX st C is closed holds A ⊆ C implies p ∈ C, (46) for A being Subset of GX ex F being Subset-Family of GX st (for C being Subset of GX holds C ∈ F iff C is closed & A ⊆ C) \ & Cl A = F, (47) for X ′ being SubSpace of GX, A being Subset of GX, A1 being Subset of X ′ st A = A1 holds Cl A1 = (Cl A) ∩ (Ω (X ′ )), (48) for A being Subset of GX holds A ⊆ Cl A, (49) for A,B being Subset of GX st A ⊆ B holds Cl A ⊆ Cl B, (50) for A,B being Subset of GX holds Cl (A ∪ B) = Cl A ∪ Cl B, (51) for A,B being Subset of GX holds Cl (A ∩ B) ⊆ (Cl A) ∩ Cl B, (52) for A being Subset of GX holds A is closed iff Cl A = A, (53) for A being Subset of GX holds A is open iff Cl (Ω (GX) \ A) = Ω (GX) \ A, (54) for A being Subset of GX, p being Point of GX holds p ∈ Cl A iff for G being Subset of GX st G is open holds p ∈ G implies A ∩ G 6= ∅(GX). 230 Beata Padlewska and Agata Darmochwal References [1] Czeslaw Byliński. Functions and their basic properties. Formalized Mathematics, 1, 1990. [2] Czeslaw Byliński. Functions from a set to a set. Formalized Mathematics, 1, 1990. [3] Beata Padlewska. Families of sets. Formalized Mathematics, 1, 1990. [4] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1, 1990. [5] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1, 1990. [6] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1, 1990. [7] Zinaida Trybulec and Halina Świe czkowska. Boolean properties of sets. Formalized ‘ Mathematics, 1, 1990. Received April 14, 1989